113 quotes found
"...the power of this line [the cycloid] to measure time."
"I believe that we do not know anything for certain, but everything probably."
"I do not mind at all that [Newton] is not a Cartesian provided he does not offer us suppositions like that of attraction."
"One finds in this subject a kind of demonstration which does not carry with it so high a degree of certainty as that employed in geometry; and which differs distinctly from the method employed by geometers in that they prove their propositions by well-established and incontrovertible principles, while here principles are tested by inferences which are derivable from them. The nature of the subject permits of no other treatment. It is possible, however, in this way to establish a probability which is little short of certainty. This is the case when the consequences of the assumed principles are in perfect accord with the observed phenomena, and especially when these verifications are numerous; but above all when one employs the hypothesis to predict new phenomena and finds his expectations realized."
"I had not thought of this regular decrease of gravity, namely that it is as the inverse square of the distance; this is a new and highly remarkable property of gravity."
"I esteem his [Newton's] understanding and subtlety highly, but I consider that they have been put to ill use in the greater part of this work, where the author studies things of little use or when he builds on the improbable principle of attraction."
"A Man that is of Copernicus’s Opinion, that this Earth of ours is a Planet, carry’d round and enlighten’d by the Sun, like the rest of the Planets, cannot but sometimes think, that it’s not improbable that the rest of the Planets have their Dress and Furniture, and perhaps their Inhabitants too as well as this Earth of ours..."
"It's evident God had no design to make a particular Enumeration in the Holy Scriptures, of all the Works of his Creation."
"These Gentlemen must be told, that they take too much upon themselves when they pretend to appoint how far and no farther Men shall go in their Searches, and to set bounds to other Mens Industry; as if they knew the Marks that God has placed to Knowledge..."
"There are many degrees of Probable, some nearer Truth than others, in the determining of which lies the chief exercise of our Judgment."
"Here we may mount from this dull Earth, and viewing it from on high, consider whether Nature has laid out all her Cost and Finery upon this small Speck of Dirt."
"We shall be less apt to admire what this World calls Great, shall nobly despise those Trifles the generality of Men set their Affections on, when we know that there are a multitude of such Earths inhabited and adorned as Well as our own."
"Now since in so many Things they... agree, what can be more probable than that in others they agree too; and that the other Planets are as beautiful and as well stock'd with Inhabitants as the Earth? Or what shadow of Reason can there be why they should not?"
"Since 'tis certain that Earth and Jupiter have their Water and Clouds, there is no reason why the other Planets should be without them. I can't say that they are exactly of the same nature with our Water; but that they should be liquid their use requires, as their beauty does that they be clear. This Water of ours, in Jupiter or Saturn, would be frozen up instantly by reason of the vast distance of the Sun. Every Planet therefore must have its own Waters of such a temper not liable to Frost."
"What a wonderful and amazing Scheme have we here of the magnificent Vastness of the Universe! So many Suns, so many Earths, and every one of them stock’d with so many Herbs, Trees and Animals, and adorn’d with so many Seas and Mountains! And how must our wonder and admiration be encreased when we consider the prodigious distance and multitude of the Stars?"
"The world is my country, to promote science is my religion."
"Like Hooke, Huygens made fundamental improvements to the clock as a time-keeping mechanism; and Hooke invented the first passable for the same purpose. ...Huygens discovered the rings of Saturn, and the formula for centrifugal force. He did important work in mechanics and optics, and one of his merits was that he made young Leibnitz enthusiastic for these subjects."
"Having converted Galileo's discovery of the isochronism of the pendulum into an accurate timepiece in 1656, Huygens had, in 1662, developed a marine variation employing a short pendulum which had subsequently been subjected to tests at sea with the aid of the English. News of the device having come to Colbert... the new director of France's economic life was determined to secure its advantages for his nation. Accordingly, Huygens was lured to Paris in 1665."
"The academicians—especially in the person of Picard—were carrying through a revolution in observational astronomy made possible by Huygens' astronomical pendulum clock, the filar micrometer perfected (if not invented) by Auzout, and the application of telescopes to large-scale graduated instruments appropriate for the measure of small angles. It was with this equipment that Picard undertook to measure the distance between two localities approximately on the meridian of Paris, to determine the differences in their latitudes, and to deduce from those results the length of degree of meridian. The eminently successful arc measure, marked by a precision thirty to forty times greater than any previously achieved..."
"Mons. Huygens found out a Method whereby the Ball of a Pendulum may be always carried along the Arch of a Cycloid."
"Huygens stated everything verbally when he was in his "geometric mode" and used [mathematical] symbols... only when he switched to his "algebraic mode." Facile mathematician that he was, he switched back and forth between the two modes as his needs changed within the same problem..."
"One of the masterpieces of seventeenth-century scientific literature was... published in 1673 under the title Horologium Oscillatorium (The Pendulum Clock). Much more than a mere description of a clock... it was in fact a treatise on the accelerated motion of a falling body, as exemplified by the bob of a pendulum clock. ...The culminating proposition is Huygen's proof that a body falling along an inverted cycloid reaches the bottom in a fixed amount of time. In other words, the cycloid is isochronous. The third section... introduces his theory of evolutes... that, among other applications, allows one to find the length of a curve. Using evolutes... he proves mathematically that the cycloidal-shaped plates will force the bob of the pendulum to move along the isochronous cycloidal path. The fourth... section... presents his theory of the compound pendulum, in which the motion of a pendulum with mass distributed along its length is compared with that of an ideal simple pendulum... The last part of the book introduces... a variant of a conical clock in which the pendulum, instead of swinging, rotates about a vertical axis... kept on an isochronous path... by the theory of evolutes."
"This [Horologium Oscillatorium] is the first modern treatise in which a physical problem is idealized by a set of parameters then analyzed mathematically. It is one of the seminal works of applied mathematics."
"Foremost, Huygens gave us precise time. His clocks were the first timekeepers to be accurate enough to be reliable in scientific experiments."
"The lasting importance of Horologium Oscillatorium stemmed more from its applied mathematics than from its pure mathematics. The next generation of mathematicians spent a great deal of time trying to find curves that satisfied specific physical properties. What other curve, if any, is a tautochrone? What curve does a hanging chain delineate? What shape does a sail take? What is the curve of fastest descent? These were the test cases for the new mathematical technique Leibniz called 'calculus.'"
"There is no direct observational evidence for the curvature [of space], the only directly observed data being the mean density and the expansion, which latter proves that the actual universe corresponds to the non-statical case. It is therefore clear that from the direct data of observation we can derive neither the sign nor that value of the curvature, and the question arises whether it is possible to represent the observed facts without introducing the curvature at all. Historically the term containing the 'cosmological constant λ' was introduced into the field equations in order to enable us to account theoretically for the existence of a finite mean density in a static universe. It now appears that in the dynamical case this end can be reached without the introduction of λ."
"It was early 1932, when Einstein and I both were at the California Institute of Technology in Pasedena, and we just decided to look for a simple relativistic model that agreed reasonably well with the known observational data, namely, the Hubble recession rate and the mean density of matter in the universe. So we took the space curvature to be zero and also the cosmological constant and the pressure term to be zero, and then it follows straightforwardly that the density is proportional to the square of the Hubble constant. It gives a value for the density that is high, but not impossibly high. That's about all there was to it. It was not an important paper, although Einstein apparently thought that it was. He was pleased to have a simple model with no cosmological constant. That's it."
"[Einstein's cosmological constant] is a name without any meaning. ...We have, in fact, not the slightest inkling of what it's real significance is. It is put in the equations in order to give the greatest possible degree of mathematical generality."
"Our own galaxy system is only one of a great many, and observations made from any of the others would show exactly the same thing: all systems are receding, not from any particular centre, but from each other: the whole system of galaxies is expanding."
"Gradually... during the second half of the nineteenth century, the uncomfortable feeling of dislike of the action at a distance, which had been so strong in Huygens and other contemporaries of Newton, but had subsided during the eighteenth century, began to emerge again, and gained strength rapidly. This was favoured by the purely mathematical transformation (which can be compared in a sense with that from the Ptolemaic to the Copernican system), replacing Newton's finite equations by the differential equations, the potential becoming the primary concept, instead of the force, which is only the gradient of the potential. These ideas, of course, arose first in the theory of electricity and magnetism or perhaps one should say in the brain of Faraday.<!--"
"In electromagnetism... the law of the inverse square had been supreme, but, as a consequence of the work of Faraday and Maxwell, it was superseded by the field. And the same change took place in the theory of gravitation. By and by the material particles, electrically charged bodies, and magnets which are the things that we actually observe come to be looked upon only as "singularities" in the field. So far this transformation from the force to the potential, from the action at a distance to the field, is only a purely mathematical operation."
"The inconsistency of first explaining matter by atoms and then explaining atoms by matter was only slowly realised, and it is only comparatively recently that we have come to see that there is nothing paradoxical in the fact that an atom or an electron, which are not matter, may have properties different from those of matter, and must be allowed to do things that a material particle could not do."
"How does it come about that we have been able to find satisfactory hypotheses to explain electricity and magnetism, light and heat, in short all other physical phenomena, but have been unsuccessful in the case of gravitation?"
"Gravitation is entirely independent of everything that influences other natural phenomena. It is not subject to absorption or refraction, no velocity of propagation has been observed. You can do whatever you please with a body, you can electrify or magnetise it, you can heat it, melt or evaporate it, decompose it chemically, its behaviour with respect to gravitation is not affected. Gravitation acts on all bodies in the same way, everywhere and always we find it in the same rigorous and simple form, which frustrates all our attempts to penetrate into its internal mechanism."
"Gravitation is not only similar to inertia in its generality, it is also measured by the same number... the mass. The inertial mass is what Newton calls the "quantity of matter": it is a measure for the resistance offered by a body to a force trying to alter its state of motion. It might be called the "passive mass." The gravitational mass, on the other hand, is a measure of the force exerted by the body in attracting other bodies. We might call it the "active" mass. The equality of active and passive, or gravitational and inertial, mass was in Newton's system a most remarkable accidental co-incidence, something like a miracle. Newton himself decidedly felt it as such, and made experiments to verify it, by swinging a pendulum with a hollow bob which could be filled with different materials. The force acting on the pendulum is proportional to its gravitational mass, the inertia to its inertial mass: the period of its swing thus depends on the ratio between these two masses. The fact that the period is always the same therefore proves that the gravitational and inertial masses are equal."
"Gradually, during the eighteenth century, physicists and philosophers had become so accustomed to Newton's law of gravitation, and to the equality of gravitational and inertial mass, that the miraculousness of it was forgotten and only an acute mind like Bessel's perceived the necessity of repeating those experiments. By the experiments of Bessel about 1830 and of Eötvös in 1909 the equality of gravitational and inertial mass has become one of the best ascertained empirical facts in physics."
"In Einstein's general theory of relativity the identity of these two coefficients, the gravitational and the inertial mass, is no longer a miracle, but a necessity, because gravitation and inertia are identical."
"There is another side to the theory of relativity. We have pointed out in the beginning how the development of science is in the direction to make it less subjective, to separate more and more in the observed facts that which belongs to the reality behind the phenomena, the absolute, from the subjective element, which is introduced by the observer, the relative. Einstein's theory is a great step in that direction. We can say that the theory of relativity is intended to remove entirely the relative and exhibit the pure absolute."
"The sequence of different positions of the same particle at different times forms a one-dimensional continuum in the four-dimensional space-time, which is called the world-line of the particle. All that physical experiments or observations can teach us refers to intersections of world-lines of different material particles, light-pulsations, etc., and how the course of the world-line is between these points of intersection is entirely irrelevant and outside the domain of physics. The system of intersecting world-lines can thus be twisted about at will, so long as no points of intersection are destroyed or created, and their order is not changed. It follows that the equations expressing the physical laws must be invariant for arbitrary transformations."
"All systems are receding, not from any particular centre, but from each other: the whole system of galactic systems is expanding."
"Let the universe have only two dimensions, and let it be the surface of an india rubber ball. It is only the surface that is the universe, not the ball itself. ...Let there be specks of dust fixed to the surface to represent the different galactic systems. If the ball is inflated, the universe expands, and these specks of dust will recede from each other, their mutual distances, measured along the surface, will increase in the same rate as the radius of the ball. An observer in any one of the specks will see all the others receding from himself, but it does not follow that he is the centre of the universe. The universe (which is the surface of the ball, not the ball itself) has no centre."
"These... are the two observational facts about our neighbourhood, which have to be accounted for by the theory: there is a finite density of matter, and there is expansion, i.e. the mutual distances are increasing, and therefore the density is decreasing."
"Since we only consider the universe on a very large scale, and make abstraction of all details and local irregularities, our universe must be homogeneous and isotropic. It follows... that the three-dimensional space of it must be what mathematicians call a space of constant curvature."
"To help us to understand three-dimensional spaces, two-dimensional analogies may be very useful... A two-dimensional space of zero curvature is a plane, say a sheet of paper. The two-dimensional space of positive curvature is a convex surface, such as the shell of an egg. It is bent away from the plane towards the same side in all directions. The curvature of the egg, however, is not constant: it is strongest at the small end. The surface of constant positive curvature is the sphere... The two-dimensional space of negative curvature is a surface that is convex in some directions and concave in others, such as the surface of a saddle or the middle part of an hour glass. Of these two-dimensional surfaces we can form a mental picture because we can view them from outside... But... a being... unable to leave the surface... could only decide of which kind his surface was by studying the properties of geometrical figures drawn on it. ...On the sheet of paper the sum of the three angles of a triangle is equal to two right angles, on the egg, or the sphere, it is larger, on the saddle it is smaller. ...The spaces of zero and negative curvature are infinite, that of positive curvature is finite. ...the inhabitant of the two-dimensional surface could determine its curvature if he were able to study very large triangles or very long straight lines. If the curvature were so minute that the sum of the angles of the largest triangle that he could measure would... differ... by an amount too small to be appreciable... then he would be unable to determine the curvature, unless he had some means of communicating with somebody living in the third dimension....our case with reference to three-dimensional space is exactly similar. ...we must study very large triangles and rays of light coming from very great distances. Thus the decision must necessarily depend on astronomical observations."
"The triangles that we can measure are not large enough... to detect the curvature. Fortunately, however, we are, in a way, able to communicate with the fourth dimension. The theory of relativity has given us an insight into the structure of the real universe: ...a four-dimensional structure. The study of the way in which the three space-dimensions are interwoven with the time-dimension affords a kind of outside point of view of the three-dimensional space... from this outside point of view we might be able to perceive the curvature of the three-dimensional world."
"Matter is actually distributed very unevenly... conglomerated into stars and galactic systems. The average density is the density that we should get if all... could be evaporated into atoms of hydrogen, or protons, and... distributed evenly over the whole of space. ...three or four protons in every cubic foot. ...a million million times less than that of the most perfect vacuum that we can produce... The universe thus consists mostly of emptiness... consider a universe without any matter at all, an empty universe, as a good approximation. But we may also take as our first approximation a universe containing... three or four protons per cubic foot. The local deviations from the average, caused by the conglomeration of matter into stars and stellar systems, are then disregarded in the grand scale model, and are only taken into account when we come to study details."
"In the beginning of 1917, two solutions of the field equations for a homogeneous isotropic universe had been found, which I... call the solutions "A" and "B." ...at that time only static solutions were looked for. It was thought that the universe must be a stable structure...In one of these solutions (B) the average density was zero, it was empty; the other one (A) had a finite density. ...In B, to get the real universe, we should have to put in a few galactic systems, in A we should have to condense the evenly distributed matter into galactic systems. The universe A... has an average density, but no expansion. It is therefore called the static universe. B, on the other hand... expands, and it could only parade in the garb of a static universe because there is nothing in it to show the expansion. B is therefore called the empty universe. Thus we had two approximations : the static universe with matter and without expansion, and the empty one without matter and with expansion.The actual universe... has both matter and expansion... In 1917... the actual value of the density was still entirely unknown, and the expansion had not yet been discovered."
"In both the solutions A and B the curvature is positive, in both three-dimensional space is finite: the universe has a definite size, we can speak of its radius, and, in the case A, of its total mass. In the case A... the density is proportional to the curvature... Thus, if we wish to have a finite density in a static universe, we must have a finite positive curvature."
"The field equations, in their most general form, contain a term multiplied by a constant, which is denoted by the Greek letter λ... sometimes called the "cosmical constant." This is a name without any meaning... We have, in fact, not the slightest inkling of what its real significance is. It is put in the equations in order to give them the greatest possible degree of mathematical generality, but, so far as its mathematical function is concerned, it is entirely undetermined: it may be positive or negative, it might also be zero."
"Purely mathematical symbols have no meaning by themselves; it is the privilege of pure mathematicians, to quote Bertrand Russell, not to know what they are talking about. ...It is the physicist, and not the mathematician, who must know what he is talking about."
"In February 1917, it was found that a static solution with a positive curvature—the solution A—was not possible without the λ. In fact the curvature is proportional to λ (in solution A, λ is equal to the curvature; in B... it is three times the curvature). Thus, at the time when we had only the two static solutions A and B, and thought that these were the only possible ones, here was a plausible physical interpretation of the meaning of λ: it was the curvature of the world, and the square root of its reciprocal, the radius of curvature, could be conceived as providing a natural unit of length."
"In the "static" universe expansion is impossible, the "empty" universe does expand. Therefore we may be tempted to consider the empty universe as the most likely approximation; and we can proceed to compute the radius of curvature of the universe, supposing it to be of the empty type, from the observed rate of expansion."
"Is the density anywhere near that corresponding to the static universe, or is it so small that we can consider the empty universe as a good approximation?"
"It is easy to compute the density of a static universe of a radius of two thousand million lightyears, and it comes out only very little larger than the observed density. The actual universe is thus very far from empty, it is, on the contrary, nearly full."
"We... come to the conclusion... that the actual universe is neither the static nor the empty one. It differs so much from both of these that neither can be used as an appropriate grand scale model. We must thus look for other solutions of the general field-equations. On account of the expansion our solution must necessarily be a non-static one, and it must have a finite density. There is only one possible static solution possessing a finite density, viz. our old friend A, but of non-static solutions with finite density there exists a great variety."
"We had become so accustomed to think of λ as an essentially positive quantity, and of a finite world with positive curvature, that the idea of investigating the possibility of solutions with negative or zero values of λ and of the curvature simply did not arise. But when this oversight was corrected, it appeared at once that in the non-static case both λ and the curvature need not be positive, but can be negative or zero quite as well."
"The way in which the universe expands is determined by the variation of this [radius of curvature] R with the time. There are three types, or families, of non-static universes... the oscillating universes, and the expanding universes of the first and of the second kind. ...each of these is a representative of a family, comprising an infinite number of members differing in size and shape. ...In the expanding family of the first kind the radius is continually increasing from... zero... In the expanding series of the second type the radius has at the initial time a certain minimum value, different for the different members of the family. [Both kinds of expanding families] become infinite after an infinite time."
"Observations give us two data, viz. the rate of expansion and the average density, and there are three unknowns: the value of λ, the sign of the curvature, and the scale of the figure, i.e. the units of R and of the time. The problem is indeterminate."
"If the curvature is small (as we know it must be, because it is imperceptible by ordinary geometric methods in our neighbourhood), then λ must be small, and if the curvature is very small, then λ must be very small. On the other hand, if λ is very small, or zero, then the curvature must be very small, and may even be zero."
"It sounds rather strange to talk of an infinite universe still expanding. If we were certain that the curvature was negative, we might still, as in the case of positive curvature, replace the phrase "the universe expands" by the equivalent one "the curvature of the universe decreases." But if the curvature is zero, and remains zero throughout, what sort of meaning are we to attach to the "expansion"? The real meaning is, of course, that the mutual distances between the galactic systems, measured in so-called natural measure, increase proportionally to a certain quantity R appearing in the equations, and varying with the time. The interpretation of R as the "radius of curvature" of the universe, though still possible if the universe has a curvature, evidently does not go down to the fundamental meaning of it."
"The manner in which time and space are bound up with each other in the four-dimensional continuum is variable. It is difficult to express this variability of the cross-connections between space and time in simple language, and different interpretations of it are possible, corresponding to different mathematical transformations of the fundamental line-element, e.g. a different choice of the variable which we interpret as "time." Perhaps the best way we can express it is by saying that the solution of the field-equations of the theory of relativity shows that there is in the universe a tendency to change its scale, which at the present time results in an expansion, but may perhaps at other times become, or have been, a shrinking. This is true of the grand scale model of the universe."
"If we put in the details, the singularities of the field, viz. the galactic systems and the stars, we find that there is... a tendency, called gravitation, to decrease the mutual distances of these "singularities." At short distances, within the confines of a galactic system, this second tendency is by far the strongest, and the galactic systems retain their size independent of the expansion or contraction of the universe..."
"A question which has long troubled astronomers and physicists is what becomes of the energy that is continually being poured out into space by the sun and the stars. To this question a complete answer is given by the new theory. It is used up, diluted, or degraded, by the expansion of the universe. ...the light travelling through the expanding universe and, so to say, trying to reach a particular star, or stellar system, which is continually receding with great velocity, is losing energy in trying to catch up with it. It is this degradation of the light, technically known as the redshift of the spectral lines, by which we become aware of the receding velocities of the extra-galactic nebulae. It can be shown that the decrease of the total amount of radiant energy in the universe by this degradation exceeds the increase by the radiation of the stars. It would not be correct, however, to conclude that the expansion is caused by the energy thus lost by the radiation..."
"It is possible to relegate the epoch of the starting of the expansion to minus infinity, e.g. by using instead of the ordinary time the logarithm of the time elapsed since the beginning. But this is only a mathematical trick. We call zero minus infinity, but that only means that we allow the universe an infinite time to get well started on its course of expansion, but it does not make the time during which anything really happens any longer."
"The "universe" is an hypothesis, like the atom, and must be allowed the freedom to have properties and to do things which would be contradictory and impossible for a finite material structure. What we observe are the stars and nebulae constituting "our neighbourhood." All that goes beyond that, in time or in space, or both, is pure extrapolation.The conclusions derived about the expanding universe depend on the assumed homogeneity and isotropy, i.e. on the hypothesis that the observed finite material density and expansion of our neighbourhood are not local phenomena, but properties of the "universe." It is not inconceivable that this hypothesis may at some future stage of the development of science have to be given up, or modified, or at least differently interpreted."
"In astronomy two characteristics are common to all data on which the solution of the great problems depends. The first is the extreme minuteness of the quantities to be measured. ...New epochs were inaugurated in the beginning of the seventeenth century by the invention of the telescope, and in the last third of the nineteenth by the discovery of photography and spectroscopy....The other characteristic is that astronomy always requires a very large number of data. ...These two characteristics of the data that the astronomer requires to build his science on make two things more necessary in astronomy than in any other science: patience and organised coöperation. ...The astronomer—each working at his own task...—is always conscious of belonging to a community, whose members, separated in space and time, nevertheless feel joined by a very real tie, almost of kinship. ...whatever his special work may be it is always a link in a chain, which derives its value from the fact that there is another link to the left and one to the right of it. It is the chain that is important, not the separate links."
"Both the law of inertia and the law of gravitation contain a numerical factor or a constant belonging to matter, which is called mass. We have thus two definitions of mass; one by the law of inertia: mass is the ratio between force and acceleration. We may call the mass thus defined the inertial or passive mass, as it is a measure of the resistance offered by matter to a force acting on it. The second is defined by the law of gravitation, and might be called the gravitational or active mass, being a measure of the force exerted by one material body on another. The fact that these two constants or coefficients are the same is, in Newton's system, to be considered as a most remarkable accidental coincidence and was decidedly felt as such by Newton himself. He made experiments to determine the equality of the two masses by swinging a pendulum, of which the bob was hollow and could be filled up with different materials. The force acting on the pendulum is proportional to its active mass, its inertia is proportional to its passive mass, so that the period will depend on the ratio of the passive and the active mass. Consequently the fact that the period of all these different pendulums was the same, proves that this ratio is a constant, and can be made equal to unity by a suitable choice of units, i.e., the inertial and the gravitational mass are the same. These experiments have been repeated in the nineteenth century by Bessel, and in our own times by Eötvös and Zeeman, and the identity of the inertial and the gravitational mass is one of the best ascertained empirical facts in physics-perhaps the best. It follows that the so-called fictitious forces introduced by a motion of the body of reference, such as a rotation, are indistinguishable from real forces. ...In Einstein's general theory of relativity there is also no formal theoretical difference, as there was in Newton's system. ...the equality of inertial and gravitational mass is no longer an accidental coincidence, but a necessity."
"We know by actual observation only a comparatively small part of the whole universe. I will call this "our neighborhood." Even within the confines of this province our knowledge decreases very rapidly as we get away from our own particular position in space and time. It is only within the solar system that our empirical knowledge extends to the second order of small quantities (and that only for g44 and not for the other gαβ), the first order corresponding to about 10-8. How the gαβ outside our neighborhood are, we do not know, and how they are at infinity of space or time we shall never know. Infinity is not a physical but a mathematical concept, introduced to make our equations more symmetrical and elegant. From the physical point of view everything that is outside our neighborhood is pure extrapolation, and we are entirely free to make this extrapolation as we please to suit our philosophical or aesthetical predilections—or prejudices. It is true that some of these prejudices are so deeply rooted that we can hardly avoid believing them to be above any possible suspicion of doubt, but this belief is not founded on any physical basis. One of these convictions, on which extrapolation is naturally based, is that the particular part of the universe where we happen to be, is in no way exceptional or privileged; in other words, that the universe, when considered on a large enough scale, is isotropic and homogeneous."
"De Sitter's redshift phenomenon is not caused by the Doppler effect of stars moving away. It is a property of space-time, which appears when these are forced into the bitter conditions of the empty universe with the lambda-term. ...de Sitter was the first to suggest, in 1917 when galaxies were not yet known, that one should try to find a redshift-distance relation for very remote celestial bodies. ...Even Friedmann, who five years later demonstrated the possibility of an expanding universe, failed to point out the redshift phenomenon as a property of his own model."
"In 1916 and 1917 de Sitter presented to the Royal Astronomical Society a series of three papers on "Einstein's Theory of Gravitation and its Astronomical Consequences." Because there was no communication between Germany and England during World War I, these papers were instrumental in introducing general relativity to the English scientific community, and they played an important part in the decision of Arthur Eddington and others to send expeditions to observe the solar eclipse of 1919..."
"Hubble, who knew nothing of the work of Friedmann and Lamaître, had read de Sitter, whose sometimes comic absentmindedness masked a supple and inventive mind. Stimulated by one of Einstein's papers on general relativity penned in 1916, de Sitter entered into a fruitful correspondence with its author and soon produced three lengthy papers of his own on the subject. In the third article, published in 1917, he simplified his calculations by assuming that the universe is devoid of matter, a mathematical fiction that he defended on the basis that the real universe is composed mostly of space anyway. He then conjectured that if two stationary objects were introduced into this void, light passing between them with respect to one another would be redshifted. Curiously, the redshift is not due to either the expansion of intergalactic space or the Doppler effect. Instead, it is the effect of the mysterious slowing down of time at great distances. ...the amount of reshift in de Sitter's model was directly proportional to the distance between the emitting and receiving objects, a relationship that had only to be tested by someone with a telescope powerful enough..."
"If de Sitter's solution were valid everywhere, then it would be thereby shown that the purpose which I pursued with the introduction of the λ-term has not been reached. In my opinion the general theory of relativity only forms a satisfactory system if according to it the physical qualities of space are completely determined by matter alone. Hence no gμv-field must be possible, i.e., no space-time-continuum, without matter that generates it."
"From 1916 Einstein and de Sitter corresponded extensively on exactly what kind of universe best fit the relativity equations. De Sitter initially developed a model of a spherical universe, in contrast to the cylindrical one Einstein had envisioned. De Sitter also tried to map out the shape of the spherical universe in absence of all matter. Einstein's reaction to de Sitter's model was strong and negative...de Sitter's sphere described a universe that changed in size instead of remaining nicely constant. ... Einstein saw matter—and its corresponding gravitational field—as what inherently created the shape of the universe. He cited what he dubbed "Mach's principle,"...the movements of any object ...were determined by all other bodies in the universe. ...how a body moves through space is tantamount to what shape space is, the concept of "shape" without matter, Einstein insisted, was meaningless."
"In 1913, Willem de Sitter suggested that fast-moving binary stars... could be used to measure the effect of a moving source on the speed of light. Various experiments of this sort over the past eight decades have verified that the speed of light received from a moving star is the same as that from a stationary star... a wealth of other detailed experiments have been carried out during the past century..."
"De Sitter pointed out that some... pairs of stars have orbits aligned so that each star is either coming towards us or going away from us as it goes round its orbit. If the speed of light is modified by the motion of the star then 'faster' light from positions when the star is approaching the Earth could catch up with 'slower' light emitted earlier when the star was moving away from us in its orbit. There would then arise the possibility of multiple ghost images. Such ghost images have not been observed... de Sitter concluded that Einstein was right... it has been argued that one needs to be careful in accepting de Sitter's binary star argument... because of a phenomenon called 'extinction.' ...To avoid this problem, the binary experiment has... been repeated with stars emitting X-rays... experiment conclusively confirmed Einstein's postulate."
"The outstanding feature... is the possibility that the velocity-distance relation may represent the de Sitter effect, and hence that numerical data may be introduced into discussions of the general curvature of space. In the de Sitter cosmology, displacement of the spectra arises from two sources, an apparent slowing down of atomic vibrations and a general tendency of material particles to scatter. The latter involves an acceleration and hence introduces the element of time. The relative importance of these two effects should determine the form of the relation between distances and velocities; and in this connection it may be emphasized that the linear relation found in the present discussion is a first approximation representing a restricted range in distance."
"If anything, de Sitter disliked Milne's program even more than Eddington's. In this dislike, he was soon joined by Herbert Dingle. ...de Sitter came to good knowledge of the genuine, actual, empirical basis of astronomical science. Secondly, reducing the observational data from the Jovian satellites practiced the young astronomer's mathematical skills in thorough and useful ways. Such a powerful combination of two skills—empirical, observational skill plus theoretical, mathematical skill—was lacking in the colleagues with whom de Sitter would later practice cosmology—with the interesting exception of Dingle."
"In "Kosmos", de Sitter shows how steps forward in the development of science can be associated with scientists, who considered the Universe from a non-static, dynamic point of view. Charles Darwin is cited as one of the examples. ...From early on, de Sitter was actively engaged in the debate between Einstein, LeMaître, Eddington and others in the significance of the General Theory of Relativity for cosmology. De Sitter found in his preferred solutions of Einstein's field equations that the Universe was expanding and the relative velocity V of recession was expected to be in proportion to the distance r. De Sitter was searching for observational evidence of such a relative recession and turned to Kapteyn for advice on the feasibility..."
"In 1917 de Sitter showed that Einstein's field equations could be solved by a model that was completely empty apart from the cosmological constant—i.e. a model with no matter whatsoever, just dark energy. This was the first model of an expanding universe. although this was unclear at the time. The whole principle of general relativity was to write equations for physics that were valid for all observers, independently of the coordinates used. But this means that the same solution can be written in various different ways... Thus de Sitter viewed his solution as static, but with a tendency for the rate of ticking clocks to depend on position. This phenomenon was already familiar in the form of gravitational time dilation... so it is understandable that the de Sitter effect was viewed in the same way. It took a while before it was proved (by Weyl, in 1923) that the prediction was of a redshifting of spectral lines that increased linearly with distance (i.e. Hubble's law). ..."
"The theorists of the time were still debating whether the universe obeyed Einstein's static model, in which no red-shift is seen, or a model by Willem de Sitter in which the universe is expanding, and distant objects are red-shifted, but the universe contains no matter. ...Few astronomers were aware of the more general expanding universe solutions found by Georges Lemaître, and, especially, Alexander Friedmann, who had found in 1922-24 a complete set of solutions for Einstein's equations for the cosmological problem. Following Hubble's paper it was quickly realized that the expanding universe solutions were the ones that were needed to understand his observations."
"De Sitter made observations in 1913 of double-star systems and became the first astronomer to prove that the velocity of light had nothing to do with the velocity of its source. ...De Sitter ...published three papers entitled "On Einstein's Theory of Gravitation and Its Astronomical Consequences. The first two... explained Einstein's theories, and the third proposed his own interpretation of the astronomical consequences... de Sitter and Einstein published a joint paper in which they proposed there may be in the universe large amounts of matter that does not emit light and consequently has not been detected. This became known as "dark matter"..."
"Shortly after Einstein published his original memoir on cosmology in 1917, de Sitter constructed an alternative static world-model, which satisfied the same laws of world-gravitation. In this model, unlike Einstein's, space-time has an intrinsic structure of its own, independent of the presence of matter. ...there is ...no matter nor radiation. Nebulae... must therefore be considered as 'test particles,' having no influence on the model as a whole. ...whereas a test particle in Einstein's universe will remain at rest if it has no intitial motion, a similar particle... in de Sitter's world will immediately acquire an ever-increasing velocity of recession from the observer. ...in de Sitter's model space-time is 'hyperbolic'. There is no absolute time, and each observer will perceive a horizon at which time will appear to stand still... This phenomenon... is only apparent, like a rainbow. At any point on the (relative) horizon the time-flux experienced by an observer there will be the same as at the original observer. Thus in de Sitter's world there will be an apparent slowing-down of distant atomic vibrations, if these keep standard time. Consequently the radiation from a distant nebula will appear to be shifted toward the red, due to an increase in wave length corresponding to the decrease in vibrational frequency. This effect... will be supplemented by the Doppler effect, due to the relative recession of the nebula regarded as a test particle."
"It is clear that at best de Sitter's world, like Einstein's, can be regarded only as a limiting form of the real world. In the Einstein world there is the greatest possible concentration of matter without motion. In the de Sitter world there is motion but no matter. However, with... a special hypothesis due to Weyl... a nebula introduced into the de Sitter world will exhibit a red-shift proportional to its distance, similar to Hubble's observational law."
"In 1932, Einstein and de Sitter, in a joint memoir, argued that the original objections of 1917 to a world model of finite density in Euclidean space no longer applied if space could be regarded as expanding. ...Einstein and de Sitter therefore constructed a homogeneous world-model of finite density, subject to the field equations of General Relativity in expanding Euclidean space. They found... their model predicted a smoothed-out density of... 4 x 10-28 grammes per cubic centimetre. Although... "perhaps on the high side... we must conclude... it is possible to represent the facts without assuming a curvature of three-dimensional space.""
"He [de Sitter] reminds us that in Einstein's paper of November 1915 the Λ term did not appear at all. Hence, Λ had obviously the value zero. Cosmologically speaking, this implies that the curvature of the Universe is proportional to Λ. In 1917, two solutions of Einstein's field equations for a homogeneous, isotropic universe were offered. ...in A the universe has a finite density, whereas in B, the average density is zero ...We are confronted with a static unverse containing matter and having no expansion on the one hand, and on the other an empty universe void of matter and expanding... when these two solutions were constructed, the expansion of the universe had not yet been discovered."
"De Sitter proposed three types of nonstatic universes: the oscillating universes and the expanding universes of the first or second kiind. The main characteristic of the expanding "family" of the first kiind is that the radius is continually increasing from a definite initial time when it had the value zero. The universe becomes infinitely large after an infinite time. In the second kind... the radius possesses at the initial time a definite minimum value... in the Einstein model... the cosmological constant is supposed to be equal to the reciprocal of R2, whereas de Sitter computed for his interpretation the constant to be equal to 3/R2. Whitrow correctly points out the significant fact that in special relativity the cosmological constant is omitted..."
"The combined effects of the stars and of Jupiter appear to determine the main statistical features of the orbits of comets."
"From a score of well-observed original orbits it is shown that the "new" long-period comets generally come from regions between about 50000 and 150000 A.U. distance. The sun must be surrounded by a general cloud of comets with a radius of this order, containing about 1011 comets of observable size; the total mass of the cloud is estimated to be of the order of 1/10 to 1/100 of that of the earth. Through the actlllllllllthis cloud into the vicinity of the sun."
"The article indicates how three facts concerning the long-period comets, which hitherto were not well understood, namely the random distribution of orbital planes and of perihelions, and the preponderance of nearly-parabolic orbits, may be considered as necessary consequences of the perturbations acting on the comets."
"The distribution of galaxies is clumpy on all scales, ranging from binaries, triples, and multiples, through groups containing a few and a hundred galaxies, to rich clusters with thousands of members, masses up to 1015±1M☉, and diameters of the order of 10 Mpc. The latter have been discussed by Neta A. Bahcall in a preceding volume of this series. The clusters are at the end of the scale of the more or less regular structures in the Universe."
"There exists also another type of galaxy agglomeration. These have diameters between roughly one and ten times those of large clusters, but have much smaller densitities; they are usually irregular, with patchy density variations and no central concentration. The larger and most conspicuous of these agglomerations may contain several clusters, which explains why they have been given the name "superclusters." In their longer dimensions, crossing times exceed the age of the Universe. They are thus unrelaxed. Unrelaxed appearance, together with large size, might be taken as a definition of superclusters."
"Soon after he joined the Leiden faculty in 1926, Dr. Oort obtained the first direct observational evidence of the Milky Way's rotation. Through complex complex calculations, he measured the relative velocities of stars moving with the rotating galaxy and deduced their distances from the galactic center. In this way, he determined that the Sun and its planets were not even close to the galaxy's center but 30,000 light years away, in the galactic hinterland, traveling an orbit of that center that takes 300 million years to complete."
"The Russian economic system is state capitalism, there called state-socialism or even communism, with production directed by a state bureaucracy under the leadership of the Communist Party. The state officials, forming the new ruling class, have the disposal over the product, hence over the surplus-value, whereas the workers receive wages only, thus forming an exploited class."
"In the present and coming times, now the Europe is devastated and mankind is impoverished by world war, it impends upon the workers of the world to organize industry, in order to free themselves from want and exploitation."
"In the present time production is dominated by capital. The capitalist, possessor of money, founded the factory, bought the machines and the raw materials, hires the workers and makes them produce goods that can be sold."
"The capitalist is not driven by the desire to provide his fellow-men with the necessities of life; he is driven by the necessity of making money."
"Owners are the shareholders, living outside the process of production, idling in distant countryhouse and maybe gambling at the exchange. A shareholder has no direct connection with the work. His property does not consist in tools of him to work; with his property consists simply in pieces of paper, in shares of enterprises of which he does not even know the hereabouts. His function in society is that of a parasite. His ownership does not mean that he commands and directs the machines; this is the sole right of the director. It means only that he may claim a certain amount of money without having to work for it. The property in hand, his shares, are certificates showing his right - guaranteed by law and government, by courts and police - to participate in the profits; titles of companionship in that large Society for Exploitation of the World, that is capitalism."
"Labor is a social process. Each enterprise is part of the productive body of society. The total social production is formed by their connection and collaboration. Like the cells that constitute a living organism, they cannot exist isolated and cut off from the body. So the organization of the work inside the shop is only one-half of the task of the workers. Over it, a still more important task, stands the joining of the separate enterprises, their combination into a social organization."
"The sense of right and wrong, the consciousness of justice in men, is not accidental . It grows up, irresistibly, by nature, out of what they experience as the fundamental conditions of their life. Society must live; so the relations of men must be regulated in such a way - it is this that law provides for - that the production of life-necessities may go on unimpeded. Right is what is essentially good and necessary for life. Not only useful for the moment, but needed generally good and necessary for life. Not only useful for the moment, but needed generally; not for the life. Not only useful for the moment, but needed generally; not for the life of single individuals, but for people at large, for the community; not for personal or temporal interests, but for the common and lasting weal. If the life-conditions change, if the system of production develops into new forms, the relations between men change, their feeling of what is right or new forms, the relations between men change, their feelings of what is right or wrong changes with them, and the law has to be altered."
"Public ownership is a middle-class program of a modernized and disguised form of capitalism. Common ownership by the producers can be the only goal of the working class."
"Private ownership, originally a means to give everybody the possibility of productive work, now has turned into the means to prevent the workers from the free use of the instruments of production. Originally a means to ascertain to the workers the fruits of their labor, it now turned into a means to deprive the workers of the fruits of their labor for the benefit of a class of useless parasites."
"Collaboration of equal companions replaces the command of masters and the obedience of servants. The sense of duty, the devotion to the community, the praise or blame of the comrades according to efforts and achievements, as incentives take the place of fear for hunger and perpetual risk of losing the job. Instead of the passive utensils and victims of capital, the workers are now the self-reliant masters and organizers of production, exalted by the proud feeling of being active co-operators in the rise of a new humanity."
"In the production under common ownership the bookkeeping is a public matter; it lies open to all. The workers have always a complete view of the course of the whole process. Only in this way they are able to discuss matters in the sectional assemblies and in the shop-committees, and to decide on what has to be done. The numerical results are made visible, moreover, by statistical tables, by graphs and pictures that display the situation at a glance. This information is not restricted to the personnel of the shop; it is a public matter, open to all outsiders. Every shop is only a member in the social production, and also the connection of its doings with the work outside is expressed in the book-keeping. Thus insight in the production going on in every enterprise is a piece of common knowledge for all the producers."
"The Workers' Councils are the form of self-government which in the times to come will replace the forms of government of the old world. Of course not for all future; none such form is for eternity. When life and work in community are natural habit, when mankind entirely controls its own life, necessity gives way to freedom and the strict rules of justice established before dissolve into spontaneous behavior. Workers' councils are the form of organization during the transition period in which the working class is fighting for dominance, is destroying capitalism and is organizing social production. In order to know their true character it will be expedient to compare them with the existing forms of organization and government as fixed by custom as self-evident in the minds of the people."
"Entirely different is the organization of common production by means of workers' councils. Social production is not divided up into a number of separate enterprises each the restricted life-task of one person or group; now it forms one connected entirety, object of care for the entirety of workers, occupying their minds as the common task of all. The general regulation is not an accessory matter, left to a small group of specialists; it is the principal matter, demanding the attention of all in conjunction. There is no separation between politics and economy as life activities of a body of specialists and of the bulk of producers. For the one community of producers politics and economy have now coalesced into the unity of general regulation and practical productive labor. Their entirety is the essential object for all."
"As an instrument of fight for the working class against capital the trade unions are losing their importance. But the fight itself cannot cease. The depressing tendencies grow stronger under big capitalism and so the resistance of the workers must grow stronger, too. Economic crises grow more and more destructive and undermine apparently secured progress. The exploitation is intensified to retard the lowering of the profit rate for the rapidly increasing capital. So again and again the workers are provoked to resistance. But against the strongly increased power of capital the old methods of fight no longer can serve. New methods are needed, and before long their beginnings present themselves. They spring up spontaneously in the wild [outlaw] strike, in the direct action."
"All who stand together in the shop, in the very same position, as direct associates, subject to the same exploitation, against the same master, stand together in common action. Their real community is the shop; personnel of the same enterprise, they form a natural union of common work, common lot and common interests. Like specters from the past the old distinctions of different membership fall back, almost forgotten in the new living reality of fellowship in common fight. The vivid consciousness of new unity enhances the enthusiasm and the feeling of power."
"Under the new circumstances discontent is universal; all the workers feel depressed under capitalist superiority; fuel for explosions has accumulated everywhere. It is not for others, it is for themselves if they join the fight. As long as they feel isolated, afraid to lose their job, uncertain what the comrades will do, without firm unity, they shrink from action. Once, however, they take up the fight, they are changed into new personalities; selfish fear recedes to the background and forth spring the forces of community, solidarity and devotion, rousing courage and perseverance. These are contagious; the example of fighting activity rouses in others, who feel in themselves the same forces awakening, the spirit of mutual and of self-confidence. Thus the wild strike as a prairie fire may spring over to other enterprises and involve ever greater masses."
"The system of production developed in Russia is State socialism. It is organized production, with the State as universal employer, master of the entire production apparatus. The workers are master of the means of production no more than under Western capitalism. They receive their wages and are exploited by the State as the only mammoth capitalist. So the name State capitalism can be applied with precisely the same meaning. The entirety of the ruling and leading bureaucracy of officials is the actual owner of the factories, the possessing class."
"The consolidation of State capitalism in Russia itself was the determining basis for the character of the Communist Party. Whilst in its foreign propaganda it continued to speak of communism and world revolution, decried capitalism, called upon the workers to join in the fight for freedom, the workers in Russia were a subjected and exploited class, living mostly in miserable working conditions, under a strong and oppressive dictatorial rule, without freedom of speech, of press, of association, more strongly enslaved than their brethren under Western capitalism. Thus an inherent falsehood must pervade politics and teachings of that party. ... The doctrine it taught under the name of Marxism was not the theory of the overthrow of highly developed capitalism by a highly developed working class, but its caricature. ... The Communist Party did not intend to make the workers independent fighters capable by their force of insight themselves to build their new world, but to make them obedient followers ready to put the party into power."
"Wars always were contests of similar competing powers, and the deepest structure of society remained the same, whether one won or the other. Contests of classes are fights for new principles, and the victory of the rising class transfers the society to a higher stage of development. Hence, compared with real war, the moral forces are of a superior kind: voluntary devoted collaboration instead of blind obedience, faith to ideals instead of fidelity to commanders, love for the class companions, for humanity, instead of patriotism. Their essential practice is not armed violence, not killing, but standing steadfast, enduring, persevering, persuading, organizing; their aim is not to smash the skulls but to open the brains. Sure, armed action will also play a role in the fight of the classes; the armed violence of the masters cannot be overcome in Tolstoian fashion like by patient suffering. It must be beaten down by force; but, by force animated by deep moral conviction."
"So the light darkened that had illuminated the world; the masses that had hailed it were left in blacker night, either in discouragement turning away from the fight, or struggling along to find new and better ways. The Russian revolution first had given a mighty impulse to the fight of the working class, by its mass direct actions and by its new council forms of organization—this was expressed in the widespread rise of the communist movement all over the world. But when then the revolution settled into a new order, a new class rule, a new form of government, State capitalism under dictatorship of a new exploiting class, the Communist Party needs must assume an ambiguous character. Thus in the course of ensuing events it became most ruinous to the working class fight, that can only live and grow in the purity of clear thought, plain deeds and fair dealings. By its idle talk of world revolution it hampered the badly needed new orientation of means and aims. By fostering and teaching under the name of discipline the vice of submissiveness, the chief vice the workers must shake off, by suppressing each trace of independent critical thought, it prevented the growth of any real power of the working class. By usurping the name communism for its system of workers' exploitation and its policy of often cruel persecution of adversaries, it made this name, till then expression of lofty ideals, a byword, an object of aversion and hatred even among workers. In Germany, where the political and economic crises had brought the class antagonisms to the highest pitch, it reduced the hard class fight to a puerile skirmish of armed youths against similar nationalist bands. And when then the tide of nationalism ran high and proved strongest, large parts of them, only educated to beat down their leaders' adversaries, simply changed colours. Thus the Communist Party by its theory and practice largely contributed to prepare the victory of fascism."
"Economic and spiritual power are made an active power through organization. It binds all the different wills to unity of purpose and combines the single forces into a mighty unity of action. Its outer forms may differ and change as to circumstances, its essence is its new moral character, the solidarity, the strong community-feeling, the devotion and spirit of sacrifice, the self-imposed discipline. Organization is the life principle of the working class, the condition of liberation. A minority ruling by its strong organization can be vanquished only, and certainly will be vanquished, by organization of the majority."